feat: add properties of spacetime

HepLean/StandardModel/CliffordAlgebra.lean

@@ -0,0 +1,79 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.StandardModel.Basic
+/-!
+# The Clifford Algebra
+
+This file defines the Gamma matrices.
+
+## TODO
+
+- Prove that the algebra generated by the gamma matrices is ismorphic to the
+  Clifford algebra assocaited with spacetime.
+-
+-/
+
+namespace StandardModel
+open Complex
+
+noncomputable section diracRepresentation
+
+def γ0 : Matrix (Fin 4) (Fin 4) ℂ :=
+  ![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
+
+def γ1 : Matrix (Fin 4) (Fin 4) ℂ :=
+  ![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]
+
+def γ2 : Matrix (Fin 4) (Fin 4) ℂ :=
+  ![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]
+
+def γ3 : Matrix (Fin 4) (Fin 4) ℂ :=
+  ![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]
+
+def γ5 : Matrix (Fin 4) (Fin 4) ℂ := I • (γ0 * γ1 * γ2 * γ3)
+
+@[simp]
+def γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ := ![γ0, γ1, γ2, γ3]
+
+
+namespace γ
+
+variable (μ : Fin 4)
+
+
+
+
+/-- The trace of the gamma matrices is zero. -/
+lemma trace_eq_zero (μ : Fin 4) : Matrix.trace (γ μ) = 0 := by
+  fin_cases μ
+    <;> simp [γ, γ0, γ1, γ2, γ3]
+    <;> rw [Matrix.trace, Fin.sum_univ_four]
+    <;> simp
+  any_goals rfl
+  change 0 + 0 = 0
+  simp [add_zero]
+
+
+
+@[simp]
+def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
+
+lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet := by
+  simp [γSet]
+
+def diracAlgebra : Subalgebra ℝ (Matrix (Fin 4) (Fin 4) ℂ) :=
+  Algebra.adjoin ℝ γSet
+
+lemma γSet_subset_diracAlgebra : γSet ⊆ diracAlgebra :=
+  Algebra.subset_adjoin
+
+lemma γ_in_diracAlgebra (μ : Fin 4) : γ μ ∈ diracAlgebra :=
+  γSet_subset_diracAlgebra (γ_in_γSet μ)
+
+end γ
+
+end diracRepresentation
+end StandardModel